\titlepage

\section{Autoassociative model of sparse connectivity}

\begin{frame}\frametitle{Autoassociative model of sparse connectivity}

The CA3-CA3 network include empirical (e.g connection probability) and it is inspired in the model described by Gibson and Robinson 1992.
\begin{columns}[T]
    \column{0.5\textwidth}    
    \begin{figure}
    \includegraphics[width=\textwidth]{../doc/figures/recurrent_network.pdf}
    \end{figure}

    \column{0.5\textwidth}    

    \begin{equation*}
    \mathbf{W} = \begin{pmatrix}
    0 & 1 & 0 & 0 & 0 \\
    0 & 0 & 1 & 0 & 0 \\
    0 & 0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 0 & 1 \\
    1 & 0 & 1 & 0 & 0 \\
    \end{pmatrix}
    \end{equation*}
\end{columns}
    
Assuming that $W_{ij}$ are independent random variables
\begin{equation*}
\Pr(W_{ij}) = \begin{cases}
    1, & \text{if} \ \ W_{ij} < c ,\\
    0, & \text{if} \ \ i=j  \ \ \text{or} \ \ W_{ij} \geq 1-c .
    \end{cases}
\end{equation*}
\end{frame}

\begin{frame}\frametitle{Autoassociative model of sparse connectivity}

The firing of the $n$ CA3 neurons in the network and patterns to be stored are represented by sequences of $0$ and ones.

\begin{center}
        $Z^0=\{0,1, \ldots, n\}$
\end{center}

\begin{columns}[T]
    \column{0.5\textwidth}    

    1.- Activity patterns \\
    
    \begin{equation*}
        Z^p:p=\{0,1, \ldots, m\}
    \end{equation*}


    2.- Storage: clipped Hebbian rule\\

    \begin{equation*}
        J_{ij} = W_{ij}Z^p_iZ^p_j 
    \end{equation*}

    \column{0.5\textwidth}    
    3.- Activation function\\
        \begin{equation*}
        h_i(t+1) = \frac{1}{n} \sum^n_j J_{ij}X_{j}(t)
        \end{equation*}
    4.- Recall 

\end{columns}

\end{frame}
